The Positive Capacity Region of Two-Dimensional Run Length Constrained Channels

A binary sequence satisfies a one-dimensional (d, k) constraint if every run of zeroes has length at least d and at most k. A binary two-dimensional array satisfies a (d, k) constraint if every run of zeroes, in each one of the array directions, has length at least d and at most k. Few models have been proposed in the literature to handle two dimensional data: the diamond model, the square model, the hexagonal model, and the triangular model. The constraints in the different directions might be asymmetric and hence many kind of constraints are defined depending on the number of directions in the model. For example, a two-dimensional array in the diamond model satisfies a (d1, k1, d2, k2) constraint if it satisfies the one-dimensional (d1, k1) constraint horizontally and the one-dimensional (d2, k2) constraint vertically. In this paper we examine the region in which the capacity of the constraints is zero or positive in the various models. We consider asymmetric constraints in the diamond model and symmetric constraints in the other models. In particular we provide an almost complete solution for asymmetric constraints in the diamond model.